I would like to verify the core fact that a complex square matrix is normal if and only if it is unitarily diagonalizable; previously, I have assumed that normal matrices admit spectral decompositions and would now like to establish that this is indeed true in the process of achieving my primary ...

I would like to establish some examples of families of (commutative) coherent algebras arising from distance-regular graphs and sets of conjugacy class digraphs of finite groups.

Firstly, suppose that [math]G[/math] is a distance-regular graph. Let [math]\mathcal{V}_{G}[/math] denote the vertex set of [math]G[/math], and [math]\mathcal{A} := \{ A_{n} : 0 \leq n \leq \operatorname{diam}(G) \}[/math] denote a set of distance matri...

Suppose that [math]L : V \to V[/math] is a linear operator acting on a finite-dimensional vector space [math]V[/math] over a field [math]\mathbb{F}[/math]. Additionally, suppose that the minimal polynomial of [math]L[/math] is given by [math]p(x) = \prod\limits_{q(x) \in S} \left ( q(x) \right )^{m_q} [/math], where [math]S[/math] denotes the set of irreducible factors of [math]p(x)[/math], and [math]m_q[/math] denotes the multiplicity of [math]q(x)[/math] as a factor of [math]p(x)[/math] for all [math]q(x) \in S[/math].

Then it is apparent that...